Fourier analysis of periodic Radon transforms

We study reconstruction of an unknown function from its $d$-plane Radon transform on the flat $n$-torus when $1 \leq d \leq n-1$. We prove new reconstruction formulas and stability results with respect to weighted Bessel potential norms. We solve the associated Tikhonov minimization problem on $H^s$ Sobolev spaces using the properties of the adjoint and normal operators. One of the inversion formulas implies that a compactly supported distribution on the plane with zero average is a weighted sum of its X-ray data.


Introduction
We study reconstruction of an unknown function from its d-plane Radon transform on the flat torus T n = R n /Z n when 1 ≤ d ≤ n − 1. The dplane Radon transform of a function f on T n encodes the integrals of f over all periodic d-planes. The usual d-plane Radon transform of compactly supported objects on R n can be reduced into the periodic d-plane Radon transform, but not vice versa. This was demonstrated for the geodesic Xray transform in the recent work of Ilmavirta, Koskela and Railo [10]. As general references on the Radon transforms, we point to [5,15,6,14].
Reconstruction formulas for integrable functions and a family of regularization strategies considered in this article were derived in [10] for the geodesic X-ray transform (d = 1) on T 2 . We extend these methods to the d-plane Radon transforms of higher dimensions, study new types of reconstruction formulas for distributions, and prove new stability estimates on the Bessel potential spaces. This article considers only the mathematical theory of Radon transforms on T n , whereas numerical algorithms (Torus CT) were implemented in [10,13].
Injectivity, a reconstruction method and certain stability estimates of the d-plane Radon transform on T n were proved for distributions by Ilmavirta in [7]. Our reconstruction formulas and stability estimates in this article are different than the ones in [7]. The first injectivity result for the geodesic X-ray transform on T 2 was obtained by Strichartz in [19], and generalized to T n by Abouelaz and Rouvière in [2] if the Fourier transform is 1 (Z n ). Abouelaz proved uniqueness under the same assumption for the d-plane Radon transform in [1].
The X-ray transform and tensor tomography on T n has been applied to other integral geometry problems. These examples include the broken ray transform on boxes [7], the geodesic ray transform on Lie groups [8], tensor tomography on periodic slabs [11], and the ray transforms on Minkowski tori [9]. We expect that the d-plane Radon transform on T n has applications in similar and generalized geometric problems as well, but have not studied this possibility any further.
This article is organized as follows. The main results are stated in section 1.1. We recall preliminaries and prove some basic properties in section 2. We prove new inversion formulas in section 3. We prove our stability estimates and theorems on Tikhonov regularization in section 4.
1.1. Results. We describe our results next. Here we only briefly introduce the used notation, and more details are given in subsequent sections. One can also find more details in [7,10]. Let n, d ∈ Z be such that n ≥ 2 and 1 ≤ d ≤ n − 1. We define the d-plane Radon transform of f ∈ C ∞ (T n ) as where A = {v 1 , . . . , v d } is any set of linearly independent integer vectors v i ∈ Z n . It can be shown that A spans a periodic d-plane on T n , and on the other hand, any periodic d-plane on T n has a basis of integer vectors. We can identify all periodic d-planes on T n by the elements in the Grassmannian space Gr(d, n) which is the collection of all d-dimensional subspaces of Q n . We redefine the d-plane Radon transform on T n as R d f : Gr(d, n) → C ∞ (T n ) without a loss of data. The definition of R d extends to the periodic distributions f ∈ T such that R d f (·, A) ∈ T for any A ∈ Gr(d, n). We use the shorter notations R d,A f = R d f (·, A) and X d,n = T n × Gr(d, n). More details are given in section 2.1.
Let w : Z n × Gr(d, n) → (0, ∞) be a weight function such that w(·, A) is at most of polynomial decay (17) for any fixed A ∈ Gr(d, n). If not said otherwise, then a weight w is always assumed to be of this form. The associated Fourier multipliers on distributions are denoted by F w . We denote the weighted Bessel potential space on the image side by L p,l s (X d,n ; w) where s ∈ R, p, l ∈ [1, ∞]. The usual Bessel potential spaces on T n are denoted by L p s (T n ), and H s (T n ) = L 2 s (T n ) is the fractional L 2 Sobolev space. The L p,l s (X d,n ; w) norms are l norms over Gr(d, n) of the w-weighted Bessel potential norms of L p s (T n ; w(·, A)) with A ∈ Gr(d, n). More details are given in section 2.2.
We show that L p,l s (X d,n ; w) are Banach spaces in lemma 2.1. Many of our results consider the Hilbert spaces with p = l = 2. Most of the theorems in this article would have been unreachable for R d when d < n − 1 if we do not include weights in the data spaces. We construct weights which satisfy the assumptions of our theorems in section 2.3.
Remark 1.1. If d = n − 1, then weights are not that important for the analysis of R d as demonstrated in the case of n = 2, d = 1 in [10], or for example in the special case of theorem 1.3.
Our first theorem considers the adjoint and the normal operators of R d : H s (T n ) → L 2,2 s (X d,n ; w). This generalizes [10, Proposition 11] into higher dimensions. Theorem 1.1 and corollary 1.2 are proved in section 2.4.3.
Theorem 1.1 (Adjoint and normal operators). Let s ∈ R and suppose that there exists C w > 0 such that for any k ∈ Z n . Then the adjoint of R d : H s (T n ) → L 2,2 s (X d,n ; w) is given by Theorem 1.1 gives a new inversion formula in terms of the adjoint and a Fourier multiplier. Its corollary 1.2 gives new stability estimates on H s (T n ). The stability estimates of R 1 on H s (T 2 ) were not explicitly written down in [10] but they can be found between the lines. We denote by R * ,w d the adjoint of R d associated to the weight w when the weight needs to be specified. Corollary 1.2 (Stability estimates). Suppose that the assumptions of theorem 1.1 hold, and that there exists c w > 0 such that W k ≥ c 2 w for any k ∈ Z n .
In order to prove L p s L p s type stability (iii) for more general weights in terms of the normal operator, one would have to show that F W −1 k is a bounded L p multiplier. Other stability estimates on H p s (T n ) are given in terms of R d f in proposition 4.3. These stability estimates follow from corollary 1.2 and the Sobolev inequality on T n . This method requires additional smoothness of R d f in order to control the norm of f due to the use of the Sobolev inequality.
We have proved three other new inversion formulas for R d as well. The first two inversion formulas are given in proposition 3.1 and its corollary 3.3. Proposition 3.1 generalizes the inversion formula [10, Theorem 1] into higher dimensions. Its corollary 3.3 generalizes the formula for all periodic distributions using the structure theorem. We state the third inversion formula here since we find it to be the most interesting one.

Theorem 1.3 (The third inversion formula)
. Suppose that f ∈ T . Let w : Z n × Gr(d, n) → R be a weight so that and the series is absolutely converging for any k ∈ Z n . (The weight does not have to generate a norm or have at most of polynomial decay.) Then Moreover, if f has zero average and d = n − 1, then The author is not aware of a similar formula for the inverse Radon transform in earlier literature. We emphasize that this new result implies that a clever sum of the (n − 1)-plane Radon transform data is the target function. If n = 2, this holds true for the X-ray transform of compactly supported functions on the plane R 2 . We further remark that it is easy to recover the average of a function and filter it out from R n−1 f .
Finally, we state our results on regularization. These results generalize [10, Theorems 2 and 3] into higher dimensions. Let g ∈ L 2,l r (X d,n ; w). We consider the Tikhonov minimization problem (8) arg min for any n ≥ 2, 1 ≤ d ≤ n − 1, α > 0, l = 2, and r, s, t ∈ R. We do not fix the regularity of f a priori but the space H t (T n ) will be found after solving the minimization problem for distributions in general. Let us define (9) P α w,z := 1 W k + α k 2z as a Fourier multiplier associated to a weight w, z ∈ R, and α > 0. Theorem 1.4 (Tikhonov minimization problem). Let w be a weight such that c 2 w ≤ W k ≤ C 2 w for some uniform constants c w , C w > 0. Suppose that α > 0, and s ≥ r. Then the unique minimizer of the Tikhonov minimization problem (8) with g ∈ L 2,2 r (X d,n ; w) is given by f = P α w,s−r R * d g ∈ H 2s−r (T n ). The last theorem we state in the introduction generalizes the result [10, Theorem 3] on regularization strategies to higher dimensions. Theorem 1.5 (Regularization strategy). Let w be a weight such that c 2 w ≤ W k ≤ C 2 w for some uniform constants c w , C w > 0. Suppose r, t, s, δ ∈ R are constants such that 2s + t ≥ r, δ ≥ 0, and s > 0. Let g ∈ L 2,2 t (X d,n ; w) and f ∈ H r+δ (T n ).
Then the Tikhonov regularized reconstruction operator P α w,s R * d is a regularization strategy in the sense that is an admissible choice of the regularization parameter. Moreover, if g L 2,2 t (X d,n ;w) ≤ , 0 < δ < 2s, and 0 < α ≤ c 2 w (2s/δ − 1), we have a quantitative convergence rate Remark 1.3. The optimal rate of convergence with respect to > 0 can be found by choosing the regularization parameter α( ) so that the terms on the right hand side of (11) are of the same order.
Acknowledgements. This work was supported by the Academy of Finland (Center of Excellence in Inverse Modelling and Imaging, grant numbers 284715 and 309963). The author is grateful to Joonas Ilmavirta who has shared his insight of the questions studied in the article. The author wishes to thank Giovanni Covi, Keijo Mönkkonen and Mikko Salo for their valuable comments on the manuscript and suggestions for improvements.

Periodic Radon transforms and Grassmannians.
We denote by T the set C ∞ (T n ) and T its dual space, i.e. the space of periodic distributions. Denote by G n d the set of linearly independent unordered d-tuples in Z n \ 0. We may write any element A ∈ G n d as A = {v 1 , . . . , v d }. The elements in the set G n d span all periodic d-planes on T n . Suppose that f ∈ T . We define the d-plane Radon transform of f as We remark that R d : T → T G n d , R d f : T n ×G n d → C and R d f (·, A) : T n → C. Denote the duality pairing between T and T by (·, ·). If f, g ∈ T , it follows easily from Fubini's theorem that We define the d-plane Radon transform for any f ∈ T and A ∈ G n d simply as (14) ( This is the unique continuous extension of R d (·, A) to the periodic distributions. The Fourier series coefficients of R d f (·, A) are defined as usual.
We denote the Grassmannian of d-dimensional subspaces of Q n by Gr(d, n). If A, B ∈ G n d span the same subspace of Q n , then A and B represent the same element in Gr(d, n) and R d f (·, A) = R d f (·, B) for any f ∈ T . On the other hand, for every A ∈ Gr(d, n) there existsÃ ∈ G n d that spans A. This allows one to define the Radon transform as R d f : Gr(d, n) → T without data redundancy by setting R d f (·, A) := R d f (·,Ã) whereÃ ∈ G n d spans A ∈ Gr(d, n). This connection to the Grassmannians was mentioned earlier in [7] but was not directly used.
Remark 2.1. Let us denote the projective space P n−1 := Gr(1, n). The height of P ∈ P n−1 is defined by H(P ) = gcd(p) −1 |p| ∞ using any representative p of P . The projective space P 1 and the height were used in [10] to analyze the number of projection directions required to reconstruct the Fourier series coefficients of a phantom up to a fixed radius. This question reduces to Schanuel's theorem [17] in algebraic number theory. This analysis in [10] extends to higher dimensions when d = n − 1.

2.2.
Bessel potential spaces and data spaces. We define the Bessel potential norms for any p ∈ [1, ∞] and s ∈ R by where k = (1 + |k| 2 ) 1/2 as usual (see e.g. [3]). The space The Bessel potential spaces are used as domains of R d in this work, which extends studies of the case p = 2 in [7,10].
If ω : Z n → (0, ∞) and f ∈ T , then we define the ω-weighted norms by where F ω is the Fourier multiplier of ω. We say that a weight ω : Z n → (0, ∞) is at most of polynomial decay if there exists C, N > 0 such that We next define suitable data spaces that contain ranges of R d when its domains are restricted to the Bessel potential spaces. Let us denote X d,n := T n × Gr(d, n) to keep our notation shorter. We generalize the data space given in [10] to all n ≥ 2, 1 ≤ d ≤ n − 1, and p ∈ [1, ∞], using the Grassmannians, the Bessel potential spaces and weights.
Let 1 ≤ d ≤ n − 1 and w : Z n × Gr(d, n) → (0, ∞) be a weight function such that w(·, A) is at most of polynomial decay for any fixed A ∈ Gr(d, n). We always assume in this work that the weight is at most of polynomial decay. We say that a (generalized) function g : is finite and g(·, A) ∈ T for any fixed A ∈ Gr(d, n). Similarly, if l = ∞, we define In the above definition, one can replace Gr(d, n) by any countable set Y (cf. lemma 2.1).
If p, l = 2, then the norm is generated by the inner product which makes L 2,2 s (X d,n ; w) a Hilbert space. We prove that the spaces L p,l s (X d,n ; w) are Banach spaces in lemma 2.1. We emphasize that a weight does not have to have uniform coefficients for its at most of polynomial decay with respect to Gr(d, n).
There is a connection to the norms used in [10]. Let w be any weight such that A∈Gr(1,2) w(0, A) 2 = 1, and w(k, A) ≡ 1 if k = 0. Now the results in [10] follow from the results of this article using the norm L 2,2 s (X 1,2 ; w) as the image side spaces in [10] are contained in L 2,2 s (X 1,2 ; w). Yet another norm was used for the stability estimates in [7]. In the cases d = n − 1 and l = ∞, our analysis of R d would not require weights, and can be performed similarly to [7,10]. The analysis of R d | L p s (T n ) has not been done before if p = 2. The Bessel potential norms on the domain side are used to understand better the mapping properties of R d .
We state and prove the following lemma for the sake of completeness. We remark that without the decay condition on weights these weighted spaces would not be complete.
weight that is at most of polynomial decay for any fixed y ∈ Y . Suppose that s ∈ R, p, l ∈ [1, ∞], and n ≥ 1. Then L p,l s (T n × Y ; w) is a Banach space. In particular, L 2,2 s (T n × Y ; w) is a Hilbert space.
Proof. Suppose that 1 ≤ l < ∞. (If l = ∞, the proof is similar.) We first show that L p,l s (T n × Y ; w) is a vector space. Let c ∈ C and f, g ∈ L p,l s (T n × Y ; w). We have trivially that The Minkowski and triangle inequalities imply This shows that L p,l s (T n × Y ; w) is a vector subspace of all collections of distributions {f (·, y)} y∈Y with f (·, y) ∈ T .
We show next that L p,l s (T n ×Y ; w) is a complete space. Let f i ∈ L p,l s (T n × Y ; w) be a Cauchy sequence. It follows from the definition of the norm in L p,l s (T n × Y ; w) that f i (·, y) ∈ L p s (T n ; w(·, y)) is a Cauchy sequence for any y ∈ Y . Suppose that each L p s (T n ; w(·, y)) is complete. It follows that f i (·, y) → f y ∈ L p s (T n ; w(·, y)) as i → ∞. This implies that there exists a limit of f i in L p,l s (T n × Y ; w) by standard arguments.
Let us prove that L p s (T n ; w(·, y)) is complete for any y ∈ Y . Take a Cauchy sequence f i ∈ L p s (T n ; w(·, y)). Now it follows that the functions are in L p (T n ) and form a Cauchy sequence. Therefore lim i→∞ g i =: g exists. We claim that the distribution defined on the Fourier side asf (k) := g(k) k s w(k,y) is the limit of f i in L p s (T n ; w(·, y)). We need to show two things, that f ∈ T and f i − f L p s (T n ;w(·,y)) → 0 as i → ∞. We have that It is enough that the Fourier coefficients of f have polynomial growth by the structure theorem of periodic distributions [18,Chapter 3 This shows that f ∈ T .
Remark 2.2. One uses the fact that weights have at most of polynomial decay only to show that the limits of Cauchy sequences are in T . One could also allow more rapid decay for weights but in that case, the objects of the completion would not be distributions but ultra-distributions [18]. In the analysis of R d , such generality seems to be unnecessary and our assumptions avoid this.

On constructions of weights.
In this section, we discuss how to construct weights that satisfy the assumptions of our theorems. The weights of this paper are of the form w : Z n × Gr(d, n) → (0, ∞) with the following properties.
(i) For any A ∈ Gr(d, n) there exists C, N > 0 such that w(k, A) ≥ C k −N for every k ∈ Z n . (ii) There exists C > 0 such that W k < C for every k ∈ Z n where W k = A∈Ω k w(k, A) 2 and Ω k = { A ∈ Gr(d, n) ; k⊥A }. (iii) There exists c > 0 such that c < W k for every k ∈ Z n . The property (i) is assumed for any weight in this article to guarantee that L p,l s (X d,n ; w) are Banach spaces. The property (ii) is assumed for most of the weights to guarantee that R d : L p s (T n ) → L p,l s (X d,n ; w) is continuous (with some restrictions if p, l = 2). The property (iii) is additionally assumed to prove the stability estimates and the theorems on regularization.
First of all, it is very easy to construct weights that satisfy (i) alone. It is not hard to construct weights that satisfy (i) and (ii). Since the set Gr(d, n) is countable, we may write it with an enumeration ϕ : Gr(d, n) → N. For example, we construct a weight w(k, A) = 2 −ϕ(A) k −N with large enough N > 0 chosen such that k∈Z n k −2N < ∞. Then A∈Gr(d,n) k∈Z n w(k, A) 2 < C for some C > 0. This shows that both conditions (i) and (ii) hold.
We give next a nontrivial example of a weight satisfying (ii) and (iii) but not (necessarily) (i). Let ϕ k : Ω k → N be an enumeration. Let Q := { (k, A) ∈ Z n ×Gr(d, n) ; A ∈ Ω k }. For any (k, A) ∈ Q, we define the weight The problem gets more difficult if the all three conditions must be satisfied at the same time. We solve this problem now by combining ideas from the both constructions above. We make a proposition about a concrete example, and more general methods are summarized in remarks 2.3 and 2.4. Proposition 2.2. Let ϕ k : Ω k → N be an enumeration for any k ∈ Z n , and let ϕ : satisfies the properties (i), (ii) and (iii).
Proof. Using the definition (27) and positivity of the involved functions, we have that This shows (iii). Suppose that (k, A) ∈ Q. We use to estimate W k from above. The formula (29) gives Since k −2N ≤ 1 and h(k) ≤ b for any k ∈ Z n , we obtain that W k ≤ This shows (ii). Using the definition (27) and positivity of the involved functions, we can directly estimate that This shows that w(·, A) is at most of polynomial decay (i).
Remark 2.3. Proposition 2.2 generalizes for w(k, A)| Q = h(k)ψ(k, A) + g(k)ω(A) with the conditions that h(k) is bounded from above and below, g(k) has at most polynomial decay and is bounded above, the sums of ω(A) 2 over Ω k are uniformly bounded from above, and the sums of ψ(k, A) 2 over Ω k are uniformly bounded from below and above.
Remark 2.4. If a weight w satisfies the conditions (i) and (ii), then it can be normalized asw(k, The normalized weightw has the property thatW k = 1 for any k ∈ Z n . Moreover, since w(k, A) is at most of polynomial decay and √ W k ≤ C for some C > 0, it follows thatw is at most of polynomial decay.
We can construct weights that satisfty the assumptions of theorem 1.3 by defining w(k, A) = 2 −ϕ k (A) for any (k, A) ∈ Q and w(k, A) = 1 if (k, A) / ∈ Q. If d < n − 1, then A∈Ω k w(k, A) = 1 for any k ∈ Z n , and the series A∈Ω k w(k, A) are absolutely convergent. 2.4. Basic properties of periodic Radon transforms. In this section, we state and prove some basic properties of R d . Some of these properties were used earlier in the special cases in [7,10]. We have chosen to include most of the proofs here for completeness.

Periodic Radon transforms for integrable functions. Let
where the formula is defined for a.e. x ∈ T n . We lighten our notation by denoting the corresponding linear combinations by T · A = t 1 v 1 + · · · + t d v d with respect to the enumeration of A. The following basic properties are valid.
Lemma 2.3. Suppose that f ∈ L 1 (T n ) and A ∈ G n d . Then R d,A f can be defined by the formula (32) for a.e. x ∈ T n . Moreover, (i) this definition coincides with the distributional definition: for every We postpone the proof of lemma 2.3 for a while. We remark that lemma 2.3 is a simple generalization of [10, Lemma 7], which was stated in [10] without a proof. We need to first introduce some useful notations.
Let q = n−d and V be the linear subspace of R n spanned by A. Now there exist distinct unit vectors e 1 A , . . . , e q A ∈ R n along the positive coordinate axes, {e 1 , . . . , e n }, such that e i A / ∈ V and E A := {v 1 , . . . , v d , e 1 A , . . . , e q A } spans R n . We define ϕ A : [0, 1] n → R n by the formula (33) ϕ A (t 1 , . . . , t q , s 1 , . . . , s d ) = t 1 e 1 A + · · · + t q e q A + s 1 v 1 + · · · + s d v d .
Remark 2.5. These coordinates are not unique, but we suppose that we have fixed some e 1 A , . . . , e q A for every A ∈ G n d . The specific choice is not important in our method.
Next we discuss some elementary properties of the coordinates ϕ A . The image of ϕ A is an n-parallelepiped when interpreted in R n . A simple calculation shows that |det(Dϕ A )| = |det(v 1 , . . . , v n )| ∈ Z + , which is also equal to the volume of the n-parallelepiped spanned by E A . The corners of the parallelepiped, ϕ A (T, S) with T ∈ {0, 1} q , S ∈ {0, 1} d , have integer coordinates as well. It can be argued that the coordinates (33) wrap around the torus |det(Dϕ A )| times when projected into T n , i.e. |det(Dϕ A )| = ϕ −1 A (x) for any x ∈ T n .
Let us denote the Lebesgue measure on T n by dm and on [0, 1] n by dx. We thus have the change of coordinates formula for integrals of measurable functions in the form of The formula (34), in a slightly different form, was used in the proofs given in [10]. The connection to [10] is explained with more details in remark 2.6.
Remark 2.6. If n = 2 and d = 1, then the formula |det(Dϕ A )| = v 2 1 holds. It is easy to check that the coordinates wrap v 2 1 times around T 2 . This formula is valid under the assumption that v 1 is not along the axis of e 1 , which in turn implies that v 2 1 = 0. If v 1 ||e 1 , then one chooses e 2 instead of e 1 in the definition of ϕ A . This is in-line with the formulas derived in [10]. In [10], the coordinates were scaled so that they wrap around T 2 just once and with interchanged roles of e 1 and e 2 . Now we are ready to prove lemma 2.3.
Proof of lemma 2.3. The properties (i) and (iii) follow easily from the definitions, and the proofs are thus omitted.
We show first that the mapping R d,A is well defined by the formula (32). Let0 = (0, . . . , 0) ∈ R d . We get from Fubini's theorem and the formula (34) that for all S ∈ R d . We show that R d,A f is a measurable function. Let α > 0 and define the sets We have already proved that the set X α is measurable for any α > 0. Now we get from the formula (36) that The set X α × [0, 1] d is measurable as a product of measurable sets. Since ϕ A is a smooth change of coordinates, we first find that ϕ A (X α × [0, 1] d ) is measurable, and thus R d,A f is measurable. Now we are ready to prove the property (ii). Suppose that f ∈ L p (T n ) and p ∈ [1, ∞). The formulas (34) and (36), and Hölder's inequality givê

2.4.2.
Mapping properties of periodic Radon transforms. We first recall the inversion formula in [7].
(2) in [7]). Let f ∈ T , k ∈ Z n and A ∈ Gr(d, n). Then It is evident that for every k ∈ Z n there exists A ∈ Gr(d, n) such that k⊥A, see [1, p. 11] and [7,Lemma 9]. This directly gives a reconstructive inversion procedure for R d . In section 3, we derive new inversion formulas which might provide computational advantage in practice (cf. [10] when n = 2 and d = 1). A ∈ Gr(d, n).

Lemma 2.5. Let
(i) If P : T → T acts as a Fourier multiplier (p k ) k∈Z n , then [P, Proof. (i) This is a simple application of theorem 2.4. We calculate that . We can conclude that The next lemma generalizes [10,Proposition 11] to many different directions.
(i) Let l ∈ [1, ∞). Suppose that for any A ∈ Gr(d, n) there exists C A > 0 such that w(k, A) = C A for every k⊥A. Moreover, suppose that Then the Radon transform R d : (ii) Suppose that for any A ∈ Gr(d, n) there exists C A > 0 such that w(k, A) = C A for every k⊥A. Moreover, suppose that Then the Radon transform R d : for any A ∈ Gr(d, n) by lemma 2.5. Theorem 2.4 implies that Now it follows from the triangle inequality and (47) that (ii) A calculation similar to the proof of (i) shows that (iii) We have that where the order of summation can be interchanged by non-negativity of the terms.
Remark 2.7. If d = n − 1, then the only restriction on w in the case of (iii) is A∈Gr(n−1,n) w(0, A) 2 < ∞. This follows since each A ∈ Gr(n − 1, n) has a unique normal direction.
2.4.3. Adjoint and normal operators. Next, we study the adjoint and normal operators of R d when the image side is equipped with the Hilbert space L 2,2 s (X d,n ; w) satisfying the assumptions (iii) of lemma 2.6. This generalizes the considerations in [10, Section 2.4] into higher dimensions and for any 1 ≤ d ≤ n − 1.
Proof of theorem 1.1. Let f ∈ H s (T n ) and g ∈ L 2,2 s (X d,n ; w). Using the definition of the inner product (20), we get where we can interchange the order of the summation by the Cauchy-Schwarz inequality as it implies that the series is absolutely convergent. We have that by the formula for the adjoint and theorem 2.4.
We prove corollary 1.2 on inversion formulas and stability estimates next.
Proof of corollary 1.2. (i) We first calculate that for any g ∈ L 2,2 s (X d,n ; w). Hölder's inequality for the sequences w(k, A) and w(k, A)ĝ(k, A) over A ∈ Ω k gives that after a rearrangement of the series. We can conclude from the formulas (53), (54) and (55) that s (X d,n ;w) . (ii) This is a simple calculation using the formula for the normal operator: (iii) We have by remark 2.4 thatw is a weight that satisfies the assumptions of theorem 1.1 andW k = 1 for any k ∈ Z n . Therefore, the corresponding adjoint R * ,w d is well-defined, and R * ,w d R d f = f for any f ∈ T by theorem 1.1.

Inversion formulas
We have already proved one new inversion formula in corollary 1.2 for H s (T n ) functions. In this section, we prove three other inversion formulas. One of the formulas generalizes the inversion formula for R 1 on L 1 (T 2 ) proved in [10, Theorem 1 and Theorem 8]. The second inversion formula is a corollary of the first one and remains valid for any distribution. The third inversion formula takes a slightly different approach and shows that a distribution f ∈ T is a weighted sum of the data R d,A f over the set Gr(d, n). These formulas might have practical value.
Proposition 3.1 (The first inversion formula). Let A ∈ Gr(d, n) and k ∈ Z n . Suppose that f ∈ T and R d,A f ∈ L 1 (T 2 ). If k⊥A, then Proof. Fubini's theorem, theorem 2.4 and the formula (34) implies that Since k⊥A, a simple calculation shows that and lemma 2.3 implies that for a.e. T ∈ [0, 1] q . Hence, using the formulas (59) and (60), we may simplify the formula (58) into the form Remark 3.1. The proof shows that instead of choosing S = 0, we may choose any other values for the S-coordinates as well.
We immediately get the following corollary from proposition 3.1 and lemma 2.3.
Remark 3.2. One could prove corollary 3.2 directly without using lemma 2.3 and theorem 2.4 (or proposition 3.1). This proof is given for the geodesic X-ray transform in [10] and it could be adapted to this setting as well.
Recall that the structure theorem of periodic distributions [16,Theorem 2.4.5] states that for any f ∈ T there exist h ∈ C(T n ) and s ≥ 0 such that We get another corollary of proposition 3.1 and lemma 2.5.

Corollary 3.3 (The second inversion formula). Let
where R d,A h(k) can be calculated by the formula (57).
We now prove our third inversion formula stated in the introduction.

Stability estimates and regularization methods
In this section, we look at stability estimates for functions in the Bessel potential spaces when p = ∞. We also generalize the Tikhonov regularization methods developed in [10]. In the Tikhonov regularization part, we restrict our study to the functions in H s (T n ), as done in [10]. Our results on regularization are new for any 1 ≤ d ≤ n − 1 when n ≥ 3, and the stability estimates are new in any dimension.

Stability estimates and the Sobolev inequality.
Recall that in corollary 1.2 we obtained the estimate if the weight w is such that the normal operator R * d R d has a uniform lower bound 1 c 2 w as a Fourier multiplier. The condition on the weight w is that This implies stability on L p s (T n ) if p ≤ 2, as we will show later. We can reach stability estimates for p > 2 using the Sobolev inequality on T n . Theorem 4.1 (Sobolev inequality [20]). Let f ∈ T . Suppose that s > 0 and 1 < q < p < ∞ satisfy s/n ≥ q −1 − p −1 . Then for some C > 0 that does not depend on f . (i) If t ∈ R, s > 0, and 1 < q < p < ∞ satisfy s/n ≥ q −1 − p −1 , then for some C > 0 that does not depend on g. (ii) If 1 ≤ p < q ≤ ∞, then for any s ∈ R holds (70) g L p,l s (X d,n ;w) ≤ g L q,l s (X d,n ;w) . Proof. (i) We have (71) g(·, A) L p (T n ;w(·,A)) ≤ C g(·, A) L q s (T n ;w(·,A)) for any A ∈ Gr(d, n) by the Sobolev inequality where C > 0 does not depend on f , A and w. Now (69) with t = 0 follows from the definition of the norms · L q,l s (X d,n ;w) and the inequality (71). Fix any z ∈ R. Define then the functiong : Gr(d, n) → T by the formulã g(·, A) = (1 − ∆) z/2 g(·, A). Now (69) with t = 0 implies (72) g L p,l z (X d,n ;w) = g L p,l 0 (X d,n ;w) ≤ C g L q,l s (X d,n ;w) = C g L q,l z+s (X d,n ;w) . (ii) The inequality (70) can be proved similarly. Now the Sobolev inequality is replaced by the inequality f L p s (T n ) ≤ f L q s (T n ) , which holds since m(T n ) = 1 and p ≤ q. Theorem 1.1 and lemma 4.2 imply the following, slightly more general, shifted stability estimates. Proposition 4.3 (Shifted stability estimates). Let w be a weight such that c 2 w ≤ W k ≤ C 2 w for some uniform constants c w , C w > 0. Let f ∈ T , s ∈ R, and s(p, n) := n p−2 2p .
s+s(p,n) (X d,n ;w) , where C 1 , C 2 > 0 do not depend on f . If p = 1, then the first inequality of (73) holds.
. Now the first inequality follows from corollary 1.2.
Suppose additionally that 1 < p < 2. Choose s p = n 2−p 2p > 0 in the part (i) of lemma 4.2. Now it holds that for any s ∈ R.
(ii) Suppose that f ∈ T and p > 2. Choose in the Sobolev inequality (68) that q = 2. Now we can calculate that the Sobolev inequality is valid if s ≥ n p−2 2p . Let us define that s p = n p−2 . Let now s ∈ R and f ∈ L p s (T n ). We then have that Now the first inequality follows from the part (i) of the theorem. The second inequality follows from the part (ii) of lemma 4.2 since p > 2.
Remark 4.1. For any f ∈ T there exists s ≥ 0 such that f ∈ L p −s (T n ) for any p ∈ [1, ∞] by the structure theorem of periodic distributions.

4.2.
Tikhonov minimization problem. We will show that P α w,s−r R * d g is the unique minimizer of (8) when l = 2. We first analyze the regularity properties of P α w,z and P α w,s−r R * d . Then we understand which space the regularized reconstruction P α w,s−r R * d g lives in when g ∈ L 2,2 r (X d,n ; w). First of all, R * d : L 2,2 r (X d,n ; w) → H r (T n ). On the other hand, P α w,z : H r (T n ) → H r+2z (T n ) for any r, z ∈ R since W k is uniformly bounded from below. We conclude that P α w,s−r R * d : L 2,2 r (X d,n ; w) → H 2s−r (T n ).
Proof of theorem 1.4. The proof uses the same ideas as the proof of [10, Theorem 2]. We have that Since the second term of (77) is independent of f , it can be neglected in the minimization problem (8). On the other hand, We next expand the term We can conclude that a solution to the minimization problem (8) is a minimizer of (80) Hence, a minimizer of (80) must minimize for each k ∈ Z n . We expand the second term of (81), and get A∈Ω k w(k, A) 2 (f (k) −ĝ(k, A)) 2 The last term of (82) does not depend on f , so it can be neglected in the minimization. Thus, we have arrived to the minimization problem Simple calculus shows that the minimizer of (83) is (84)f (k) = A∈Ω k w(k, A) 2ĝ (k, A) W k + α k 2s−2r = F(P α w,s−r R * d g)(k).
Hence, the unique minimizer of (8) is f = P α w,s−r R * d g. The claimed regularity of f follows from the discussion preceding the proof.
Remark 4.2. If l = 2, the analysis of the Tikhonov minimization problem becomes more difficult but it might still be possible to adapt the method also in that case (when p = 2). 4.3. Regularization strategies. Let X and Y be subsets of Banach spaces and F : X → Y a continuous mapping. A family of continuous maps R α : Y → X with α ∈ (0, α 0 ], α 0 > 0, is called a regularization strategy if lim α→0 R α (F (x)) = x for any x ∈ X. A choice of regularization parameter α( ) with lim →0 α( ) = 0 is called admissible if holds for any x ∈ X [4,12]. We will show that the solution found in theorem 1.4 to the Tikhonov minimization problem (8) is an admissible regularization strategy with a quantitative stability estimate. Our proof follows that of [10,Theorem 3].
Proof of theorem 1.5. Let α > 0. Theorem 1.1 implies that To estimate the first term on the right hand side of (86), we calculate that as a Fourier multiplier. This shows that P α w,s F W k − Id H r (T n )→H r (T n ) = 1 as W k is bounded from below and above. It follows from the dominated convergence theorem that (P α w,s F W k − Id)f 2 r → 0 as α → 0 if f ∈ H r (T n ). Suppose that g L 2,2 t (X d,n ;w) ≤ . We have that R * d = R d = C w by lemma 2.6. Hence R * d g 2 H t (T n ) ≤ C 2 w 2 . This implies that P α w,s R * d g 2 H r (T n ) ≤ C 2 where the last inequality follows using −4s + 2r − 2t ≤ 0. We can conclude that This shows that choosing α = √ gives a regularization strategy. Suppose now that δ > 0. The proof of the estimate (11) is similar to that of [10]. Using the formula (87), we get that (90) P α w,s F W k − Id H r+δ (T n )→H r (T n ) = sup We can estimate the norm by defining the functions (91) F k (x) := αW −1 k x 2s−δ 1 + αW −1 k x 2s . The formula [10,Eq. (38)] implies that the maximum value of F k is (W −1 k α) δ/2s C(δ/2s) if α ≤ W k (2s/δ − 1). We see that α ≤ W k (2s/δ − 1) holds as we assumed that α ≤ c 2 w (2s/δ − 1).